metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C59⋊C8, C118.C4, C4.2D59, C2.Dic59, C236.2C2, SmallGroup(472,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C59 — C59⋊C8 |
Generators and relations for C59⋊C8
G = < a,b | a59=b8=1, bab-1=a-1 >
Smallest permutation representation of C59⋊C8
►Regular action on 472 points
Generators in S472
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59)(60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118)(119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177)(178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236)(237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295)(296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354)(355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413)(414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472)
(1 414 194 296 70 355 145 237)(2 472 195 354 71 413 146 295)(3 471 196 353 72 412 147 294)(4 470 197 352 73 411 148 293)(5 469 198 351 74 410 149 292)(6 468 199 350 75 409 150 291)(7 467 200 349 76 408 151 290)(8 466 201 348 77 407 152 289)(9 465 202 347 78 406 153 288)(10 464 203 346 79 405 154 287)(11 463 204 345 80 404 155 286)(12 462 205 344 81 403 156 285)(13 461 206 343 82 402 157 284)(14 460 207 342 83 401 158 283)(15 459 208 341 84 400 159 282)(16 458 209 340 85 399 160 281)(17 457 210 339 86 398 161 280)(18 456 211 338 87 397 162 279)(19 455 212 337 88 396 163 278)(20 454 213 336 89 395 164 277)(21 453 214 335 90 394 165 276)(22 452 215 334 91 393 166 275)(23 451 216 333 92 392 167 274)(24 450 217 332 93 391 168 273)(25 449 218 331 94 390 169 272)(26 448 219 330 95 389 170 271)(27 447 220 329 96 388 171 270)(28 446 221 328 97 387 172 269)(29 445 222 327 98 386 173 268)(30 444 223 326 99 385 174 267)(31 443 224 325 100 384 175 266)(32 442 225 324 101 383 176 265)(33 441 226 323 102 382 177 264)(34 440 227 322 103 381 119 263)(35 439 228 321 104 380 120 262)(36 438 229 320 105 379 121 261)(37 437 230 319 106 378 122 260)(38 436 231 318 107 377 123 259)(39 435 232 317 108 376 124 258)(40 434 233 316 109 375 125 257)(41 433 234 315 110 374 126 256)(42 432 235 314 111 373 127 255)(43 431 236 313 112 372 128 254)(44 430 178 312 113 371 129 253)(45 429 179 311 114 370 130 252)(46 428 180 310 115 369 131 251)(47 427 181 309 116 368 132 250)(48 426 182 308 117 367 133 249)(49 425 183 307 118 366 134 248)(50 424 184 306 60 365 135 247)(51 423 185 305 61 364 136 246)(52 422 186 304 62 363 137 245)(53 421 187 303 63 362 138 244)(54 420 188 302 64 361 139 243)(55 419 189 301 65 360 140 242)(56 418 190 300 66 359 141 241)(57 417 191 299 67 358 142 240)(58 416 192 298 68 357 143 239)(59 415 193 297 69 356 144 238)
G:=sub<Sym(472)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59)(60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118)(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177)(178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236)(237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295)(296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354)(355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413)(414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472), (1,414,194,296,70,355,145,237)(2,472,195,354,71,413,146,295)(3,471,196,353,72,412,147,294)(4,470,197,352,73,411,148,293)(5,469,198,351,74,410,149,292)(6,468,199,350,75,409,150,291)(7,467,200,349,76,408,151,290)(8,466,201,348,77,407,152,289)(9,465,202,347,78,406,153,288)(10,464,203,346,79,405,154,287)(11,463,204,345,80,404,155,286)(12,462,205,344,81,403,156,285)(13,461,206,343,82,402,157,284)(14,460,207,342,83,401,158,283)(15,459,208,341,84,400,159,282)(16,458,209,340,85,399,160,281)(17,457,210,339,86,398,161,280)(18,456,211,338,87,397,162,279)(19,455,212,337,88,396,163,278)(20,454,213,336,89,395,164,277)(21,453,214,335,90,394,165,276)(22,452,215,334,91,393,166,275)(23,451,216,333,92,392,167,274)(24,450,217,332,93,391,168,273)(25,449,218,331,94,390,169,272)(26,448,219,330,95,389,170,271)(27,447,220,329,96,388,171,270)(28,446,221,328,97,387,172,269)(29,445,222,327,98,386,173,268)(30,444,223,326,99,385,174,267)(31,443,224,325,100,384,175,266)(32,442,225,324,101,383,176,265)(33,441,226,323,102,382,177,264)(34,440,227,322,103,381,119,263)(35,439,228,321,104,380,120,262)(36,438,229,320,105,379,121,261)(37,437,230,319,106,378,122,260)(38,436,231,318,107,377,123,259)(39,435,232,317,108,376,124,258)(40,434,233,316,109,375,125,257)(41,433,234,315,110,374,126,256)(42,432,235,314,111,373,127,255)(43,431,236,313,112,372,128,254)(44,430,178,312,113,371,129,253)(45,429,179,311,114,370,130,252)(46,428,180,310,115,369,131,251)(47,427,181,309,116,368,132,250)(48,426,182,308,117,367,133,249)(49,425,183,307,118,366,134,248)(50,424,184,306,60,365,135,247)(51,423,185,305,61,364,136,246)(52,422,186,304,62,363,137,245)(53,421,187,303,63,362,138,244)(54,420,188,302,64,361,139,243)(55,419,189,301,65,360,140,242)(56,418,190,300,66,359,141,241)(57,417,191,299,67,358,142,240)(58,416,192,298,68,357,143,239)(59,415,193,297,69,356,144,238)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59)(60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118)(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177)(178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236)(237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295)(296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354)(355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413)(414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472), (1,414,194,296,70,355,145,237)(2,472,195,354,71,413,146,295)(3,471,196,353,72,412,147,294)(4,470,197,352,73,411,148,293)(5,469,198,351,74,410,149,292)(6,468,199,350,75,409,150,291)(7,467,200,349,76,408,151,290)(8,466,201,348,77,407,152,289)(9,465,202,347,78,406,153,288)(10,464,203,346,79,405,154,287)(11,463,204,345,80,404,155,286)(12,462,205,344,81,403,156,285)(13,461,206,343,82,402,157,284)(14,460,207,342,83,401,158,283)(15,459,208,341,84,400,159,282)(16,458,209,340,85,399,160,281)(17,457,210,339,86,398,161,280)(18,456,211,338,87,397,162,279)(19,455,212,337,88,396,163,278)(20,454,213,336,89,395,164,277)(21,453,214,335,90,394,165,276)(22,452,215,334,91,393,166,275)(23,451,216,333,92,392,167,274)(24,450,217,332,93,391,168,273)(25,449,218,331,94,390,169,272)(26,448,219,330,95,389,170,271)(27,447,220,329,96,388,171,270)(28,446,221,328,97,387,172,269)(29,445,222,327,98,386,173,268)(30,444,223,326,99,385,174,267)(31,443,224,325,100,384,175,266)(32,442,225,324,101,383,176,265)(33,441,226,323,102,382,177,264)(34,440,227,322,103,381,119,263)(35,439,228,321,104,380,120,262)(36,438,229,320,105,379,121,261)(37,437,230,319,106,378,122,260)(38,436,231,318,107,377,123,259)(39,435,232,317,108,376,124,258)(40,434,233,316,109,375,125,257)(41,433,234,315,110,374,126,256)(42,432,235,314,111,373,127,255)(43,431,236,313,112,372,128,254)(44,430,178,312,113,371,129,253)(45,429,179,311,114,370,130,252)(46,428,180,310,115,369,131,251)(47,427,181,309,116,368,132,250)(48,426,182,308,117,367,133,249)(49,425,183,307,118,366,134,248)(50,424,184,306,60,365,135,247)(51,423,185,305,61,364,136,246)(52,422,186,304,62,363,137,245)(53,421,187,303,63,362,138,244)(54,420,188,302,64,361,139,243)(55,419,189,301,65,360,140,242)(56,418,190,300,66,359,141,241)(57,417,191,299,67,358,142,240)(58,416,192,298,68,357,143,239)(59,415,193,297,69,356,144,238) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59),(60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118),(119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177),(178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236),(237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295),(296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354),(355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413),(414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472)], [(1,414,194,296,70,355,145,237),(2,472,195,354,71,413,146,295),(3,471,196,353,72,412,147,294),(4,470,197,352,73,411,148,293),(5,469,198,351,74,410,149,292),(6,468,199,350,75,409,150,291),(7,467,200,349,76,408,151,290),(8,466,201,348,77,407,152,289),(9,465,202,347,78,406,153,288),(10,464,203,346,79,405,154,287),(11,463,204,345,80,404,155,286),(12,462,205,344,81,403,156,285),(13,461,206,343,82,402,157,284),(14,460,207,342,83,401,158,283),(15,459,208,341,84,400,159,282),(16,458,209,340,85,399,160,281),(17,457,210,339,86,398,161,280),(18,456,211,338,87,397,162,279),(19,455,212,337,88,396,163,278),(20,454,213,336,89,395,164,277),(21,453,214,335,90,394,165,276),(22,452,215,334,91,393,166,275),(23,451,216,333,92,392,167,274),(24,450,217,332,93,391,168,273),(25,449,218,331,94,390,169,272),(26,448,219,330,95,389,170,271),(27,447,220,329,96,388,171,270),(28,446,221,328,97,387,172,269),(29,445,222,327,98,386,173,268),(30,444,223,326,99,385,174,267),(31,443,224,325,100,384,175,266),(32,442,225,324,101,383,176,265),(33,441,226,323,102,382,177,264),(34,440,227,322,103,381,119,263),(35,439,228,321,104,380,120,262),(36,438,229,320,105,379,121,261),(37,437,230,319,106,378,122,260),(38,436,231,318,107,377,123,259),(39,435,232,317,108,376,124,258),(40,434,233,316,109,375,125,257),(41,433,234,315,110,374,126,256),(42,432,235,314,111,373,127,255),(43,431,236,313,112,372,128,254),(44,430,178,312,113,371,129,253),(45,429,179,311,114,370,130,252),(46,428,180,310,115,369,131,251),(47,427,181,309,116,368,132,250),(48,426,182,308,117,367,133,249),(49,425,183,307,118,366,134,248),(50,424,184,306,60,365,135,247),(51,423,185,305,61,364,136,246),(52,422,186,304,62,363,137,245),(53,421,187,303,63,362,138,244),(54,420,188,302,64,361,139,243),(55,419,189,301,65,360,140,242),(56,418,190,300,66,359,141,241),(57,417,191,299,67,358,142,240),(58,416,192,298,68,357,143,239),(59,415,193,297,69,356,144,238)]])
124 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 59A | ··· | 59AC | 118A | ··· | 118AC | 236A | ··· | 236BF |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 59 | ··· | 59 | 118 | ··· | 118 | 236 | ··· | 236 |
| size | 1 | 1 | 1 | 1 | 59 | 59 | 59 | 59 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
124 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D59 | Dic59 | C59⋊C8 |
| kernel | C59⋊C8 | C236 | C118 | C59 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 29 | 29 | 58 |
Matrix representation of C59⋊C8 ►in GL3(𝔽1889) generated by
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1888 | 957 |
| 1804 | 0 | 0 |
| 0 | 503 | 603 |
| 0 | 279 | 1386 |
G:=sub<GL(3,GF(1889))| [1,0,0,0,0,1888,0,1,957],[1804,0,0,0,503,279,0,603,1386] >;
C59⋊C8 in GAP, Magma, Sage, TeX
C_{59}\rtimes C_8 % in TeX
G:=Group("C59:C8"); // GroupNames label
G:=SmallGroup(472,1);
// by ID
G=gap.SmallGroup(472,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-59,8,21,7427]);
// Polycyclic
G:=Group<a,b|a^59=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export